Schwarzer, 40, who played in the 1-1 World Cup qualifying draw against Japan on Tuesday, is now looking for a new club after five years at Craven Cottage.
"I've been hearing that for a while," he said of Stekelenburg's arrival. "It's no great surprise and I'll be looking for another club.
"Nothing has been sorted at the moment, the Japan game was the most important thing on the agenda so I'm worried about the Australia games now."
However, the Australian is unlikely to move back home to the A-League.
"Not at this stage, no. My family's settled and the kids are in school and I wouldn't (move) my family anyway."